OPTION PRICES AND IMPLIED VOLATILITIES:
AN EMPIRICAL ANALYSIS
Malcolm Edey
Graham Elliott
Research Discussion Paper
8902
May 1989
Research Department
Reserve Bank of Australia
The views expressed herein are those of the authors and do not necessarily
reflect those of the Reserve Bank of Australia.
ABSTRACT
This paper investigates the efficiency of Australian options markets using a
version of the Black-Scholes model.
Under the joint null hypothesis that the
pricing model is valid, and that forecasts are efficient, the implied
volatilities calculated from observed option prices should be efficient
predictors of squared changes in the prices of the underlying instruments.
This hypothesis is tested using weekly data on prices of Australian financial
futures options, and over-the-counter currency options.
The results indicate
significant forecasting biases for each of the contracts studied.
In each
case, implied volatilities appear to overpredict changes in the true
volatility of underlying prices.
Although these conclusions are conditional
on the validity of the pricing model used to calculate implied volatilities,
our evidence suggests that biases in the Black-Scholes formula are unlikely to
explain fully the apparent forecast biases.
i
TABLE OF CONTENTS
Abstract
( i)
(ii)
Table of Contents
1.
Introduction
1
2.
A Test of Forecast Efficiency
2
(a)
Futures Options
2
(b)
Currency Options
6
3.
Data Employed
7
4.
Results
8
5.
Further Results on Pricing Discrepancies
12
6.
Conclusions
17
References
18
ii
OPTION PRICES AND IMPLIED VOLATILITIES:
AN EMPIRICAL ANALYSIS
Malcolm Edey and Graham Elliott
1.
Introduction
Empirical analysis of option prices has focussed on two related but logically
distinct questions.
The first is concerned with discriminating between
alternative pricing models.
The widely used Black-Scholes model has the
attraction of being both mathematically rigorous and relatively simple to use,
since it specifies option values as a closed function of a small number of
parameters which can be readily observed or estimated.
Its validity, however,
depends on a number of restrictive assumptions concerning the stochastic
processes generating prices of the underlying assets.
In particular, it
assumes that asset prices follow diffusion processes with constant variances,
and this assumption is thought to be unrealistic in many contexts.
The
Black-Scholes model has been generalised in a number of important directions
to allow for a wider range of generating processes permitting, for example,
price discontinuities and time-varying volatilities.
A number of studies have
focussed on the performance of such models relative to Black-Scholes in
explaining observed option prices.
A second question concerns the accuracy with which market participants
estimate the parameters needed to implement the option pricing formulas.
Efficient markets theory hypothesises that the market's estimates of these
parameters may be found to be statistically optimal, in the sense that they
cannot be improved upon using any information available at the time the
expectations are formed.
This hypothesis is directly testable, conditional on
assumptions about the appropriate pricing model.
In the case of the
Black-Scholes model, for example, the parameter of prime importance is the
expected variance of the underlying asset price, and given the Black-Scholes
assumptions, market estimates of this parameter can be inferred from observed
option premiums.
Forecast efficiency can thus be tested by comparing these
implied volatilities with actual price volatilities observed over the
subsequent life of the option.
These two empirical approaches are of course complementary, each assuming one
part of the joint hypothesis in order to test the other.
The present study
falls within the second category, and is aimed specifically at testing the
efficiency of volatility expectations implied in prices of Australian futures
and currency options.
We know of no earlier study which examines these
particular options markets in Australia.
For futures options, the study uses
2.
the Black-Scholes formula as modified by Black (1976) to obtain time series
for implied volatilities;
version is used.
for currency options, the Garman-Kohlhagen (1983)
The study derives testable implications relating these
implied volatilities to subsequent price movements in the underlying
instruments.
In doing so, it follows an approach similar to that used in a
number of earlier studies using data on U.S. stock options, for example,
Schmalensee and Trippi (1978), Latane and Rendleman (1976) and Chiras and
Manaster (1978), as well as more recent work by Shastri and Tandon (1986) on
currency options.
This work has generally found evidence against the
hypothesis of forecast efficiency, although the issue remains unclear because
of the conditional nature of the hypothesis tests.
The present study aims to
provide comparable evidence using data on Australian futures and currency
options, and will also attempt to test the robustness of the statistical
results by examining whether a hypothetical trading rule, aimed at exploiting
apparent forecast inefficiencies, generates significant excess returns during
the sample period.
Section 2 of the paper derives the tests of forecast efficiency to be used in
the empirical work.
Section 3 then discusses the data used and section 4
presents the main empirical results.
Section 5 reports on an examination of
within sample excess returns using a hedged trading strategy based on the
estimated volatility predictions.
2.
{a)
Some conclusions are offered in section 6.
A Test of Forecast Efficiency
Futures Options
The Black model for pricing options on futures contracts may be written in the
following form:
c
= c(f,
x, t*, o)
(1)
p
= p(f,
x, t*, o)
( 2)
where c is the price of a call option
p is the price of a put option
f is the futures price at time t
X
is the option exercise price
is the standard deviation of the futures price
1
t* is the time to expiry.
0
3.
The model assumes that both variances and interest rates are non-stochastic,
and that the options cannot be exercised before expiry.
Recent theoretical
work by Ramaswamy and Sundaresan (1985), Schaeffer and Schwartz (1987} and
Hull and White (1987) has begun to quantify the effects on option values when
these assumptions are loosened;
generally speaking the effects appear small
when options are near the money or are relatively close to expiry.
For
example, the authors cited above compute pricing biases of the order of
between zero and 1 per cent in Black-Scholes prices for at-the-money calls
when the assumptions are violated.
This may be compared with the size of
discrepancies arising from likely errors in forecasting volatility.
As an
illustration, a 1 percentage point prediction error in estimating volatility
on a one year option with true volatility of 0.1, would produce a mispricing
of at-the-money calls of the order of 10 per cent under the Black-Scholes
formula; such a prediction error would appear quite plausible when compared
with observed historical variation in implied volatilities.
These magnitudes,
and casual observation, suggest that beliefs about volatility are likely to be
much more important in determining actual option prices than beliefs about
what is the appropriate pricing model, especially for options that are near
the money.
On this basis, we believe that an empirical focus on forecast
accuracy rather than model accuracy is not unwarranted.
It remains true,
however, that the empirical results must be interpreted as being conditional
on the assumption that Black-Scholes is a good approximation to the "true"
formula.
All of the parameters of the Black model are readily observed in historical
series apart from a, which market participants are assumed to estimate.
Given the option price, market estimates of a can therefore be inferred by
1.
Interest rates are deleted from the formula on the basis that futures
options are purchased on margin with no interest opportunity cost. The
exact formulas used are:
c
=
p
where dl =
-fN(-d ) + xN(-d )
1
2
log( fix)
a.ft*
d2 = dl - a.ft*
+
1 a.ft*
2
4.
numerically solving equation (1) or (2).
Cox and Rubinstein (1985) show that,
provided conditional variances are non-stochastic, the implied values of
o
2
taken at any time t can be interpreted as annualised conditional
variances of the log expiry price.
We can therefore write:
( 3)
2
where the notation o (t, T) denotes the implied variance observed at t for
an option expiring at T, expressed as an annual magnitude.
Et+i(log fT) - Et+i-l (log fT)
=ct+i'
2
2
(T - t)o (t, T)
=
E (I Et .)
t
+~
.
~
=
Defining
we have
T-t
I:
i=l
(4)
The cross product terms in the above expression are eliminated by the
rationality requirement that future revisions to forecasts are not correlated
with current information at any point.
Empirically, some measure of the
innovation terms will be needed, and this paper uses the assumption that
log f
.
t+~,
which implies that
A theoretical justification for the above assumption is provided by Samuelson
(1965), and strong empirical support is provided in an earlier study, Edey and
Elliott (1988), which uses the same data set as the present paper.
By leading equation (4) one period, we also have
Taking expectations at time t:
2
Et(T-(t+l))o (t+l, T)
=
5.
This can be substituted into equation (4) to give
2
(T-t)o (t, T)
E
t
{c 2
t+ 1
+
T
L
i=2
2
(ct .)}
+1
(5)
Equation (5) is a linear prediction equation relating current implied
volatility to expected outcomes on observable variables realised in the next
period.
Standard efficient markets principles can be used to derive testable
restrictions for this equation.
In particular, if the current value of o
2
is an efficient predictor, then significant predictive power should not be
added by the inclusion of further information dated t or earlier.
This suggests a test of forecast efficiency using an equation of the form
2
2
(T-(t+l))o (t+l, T) + ct+l
(T-t)~
2
+ B(T-t)o (t, T) + (T-t)yzt + vt+l (6)
where Zt is a vector representing information observable at t.
The
additional extraneous regressors are premultiplied by (T-t) to ensure that
they always have the same order of magnitude as the quantity being forecast.
In the tests reported in section 4, Zt is taken to include current and past
2
values of ct.
Under the null hypothesis,
~
= y = 0, and
B
= 1.
The interpretation of equation (6) is fairly straightforward.
Under the
efficient markets hypothesis, the current estimate of volatility remaining
over the life of the option should be an optimal predictor of realised
volatility, expressed as the appropriately weighted sum of next period's
estimate and the realised squared price innovation in the next period.
Although equation (6) is similar in spirit to equations tested by other
researchers in this context, the exact linear predictive relationship used
here has not to our knowledge been previously noted.
As an aside we note that a possible problem with the proposed test is that if
the o
2
time series is nonstationary, the estimated B coefficient would be
biased towards one, since the nonstationary magnitude appears on both sides of
the regression equation.
that the o
2
However the statistical results strongly suggest
series are stationary.
6
(b)
0
Currency Options
Although options on currency futures are traded on the Sydney Futures
Exchange, the most active currency options market in Australia is in
over-the-counter options, traded mainly in the interbank market.
There is an
important difference in the expiry date conventions as between the two
markets.
In futures options, only a small number of standard expiry dates are
used (coinciding with futures expiry dates), so that a time series of data can
be used to obtain a series of observations pertaining to the same expiry
date.
In over-the-counter options, the main indicator rates are for standard
periods of time ahead of the trading date;
thus a time series of data will
show a series of options with equal time to expiry.
In this case the algebra
for deriving a testable equation is somewhat simplified because there is no
need to "accumulate" variances in the way used above.
The equivalent of
equation (3) in this case is
2
ko (t, t+k)
k
2
E ( }:, ct . )
t i=l
+1
(7a)
(7b)
leading to the regression equations
k
I
i=l
ct+i
)2
2
= a. + Bko ( t, t+k) + yZt+ v t+k
(Sa)
or
(8b)
The difference between the two regression specifications is that the second
imposes the zero restrictions on the expected value of the cross product
terms, implied under the null hypothesis.
This effectively removes one source
of noise from the left hand side, and should result in improved efficiency in
estimation.
Equation (8b) is therefore used in the empirical work.
To
implement the equation, it is assumed that
where S is the spot exchange rate, thus assuming that the log of the spot
exchange rate follows a random walk.
Implied volatilities are obtained using
the Garman-Kohlhagen (1983) version of the Black-Scholes model, developed for
7.
pricing currency options.
Because the error terms in equations {8a and 8b)
contain overlapping forecast errors, the method of Hansen and Hedrick {1980)
is used to correct the estimated standard errors for the resulting serial
correlation.
3.
Data Employed
Our data set on futures options comprises weekly observations on the three
major contracts traded on the Sydney Futures Exchange {in 90-day bills,
10-year bonds, and the Share Price Index), with the data period running from
the inception of trading in each option up to the end of May 1988, using
Wednesday observations.
At each date a single put and call option are
selected, in each case choosing the option most heavily traded on that day.
The reason for using this criterion is that most of the SFE options quoted at
a point in time are not actively traded, and the study aims to derive its
results only from those options which are most liquid.
Also, volatilities
derived from puts and calls are tested separately because there is some
evidence {see for example Brenner, Courtadon and Subrahmanyam {1985)) of
significantly different results for the two types of options.
The approach
used here thus differs slightly from some earlier work which uses weighted
averages of implied volatilities taken from several options at each date.
The options used in the study are always written on the nearest maturing
futures contract {or the following one, when the maturity date is very near)
and have the same expiry date as their associated futures contracts.
thus range between zero and about 90 days to maturity.
They
On one observation
each 90 days, it is not possible to observe the relationship described in
equation {6), since the consecutive weekly observations will be for options
with differing maturity dates.
"New contract" dummies at these points are
therefore added to the equation for the purposes of estimation.
These ensure
that observations coinciding with contract changeovers have no influence on
the statistical results.
A helpful consequence of selecting only the most
heavily traded options is that these are generally the available options that
are nearest the money.
Pricing biases arising from violation of the
Black-Scholes assumptions should therefore be minimised in this data set,
since it is generally thought that the Black-Scholes formula performs worst
for options which are not close to the money.
The currency options data are weekly Wednesday quotes for at-the-money put and
call options on the $A/US$ exchange rate, of one month maturity.
period is from December 1984 to December 1987.
The data
8.
4.
Results
The implied volatilities calculated from the data exhibit considerable
variation through time, as can be seen in Figures 1 to 4.
A minimal
requirement for this behaviour to be consistent with rational forecasting is
the existence of conditional heteroskedasticity in the time series for future
prices.
A useful means of testing for this in an atheoretical context is to
specify a generalised ARCH process of the form:
2
2
a + Lb.et
. + cot
1
-1
where the e's represent price innovations, and are observable.
The equation
is atheoretical because it makes no attempt to match up the time horizon
covered by the option implied volatilities with the timing of subsequent
innovations.
Nevertheless the test is offered as a preliminary indication of
the information contained in option prices.
If the b and c coefficients are
jointly zero, the prima facie conclusion would be that variances are constant,
and that any changes in implied volatilities would in reality be forecasting
changes in variance which do not subsequently occur.
In fact the results,
summarised in Table 1, provide fairly robust evidence in favour of significant
heteroskedasticity of the underlying prices, so that at least this weak form
of efficiency is not rejected by the data.
usually significant.
Also, the implied volatilities are
A reassuring aspect of this result is that the
B
coefficient in the main regression equation (6), cannot then simply be
interpreted as representing the ability of o! to predict o!+l"
The implied
volatility o! does contain at least some information useful for forecasting
2
the expected value of et+l"
The main sets of results involve direct testing of equations (6) and (8b), and
are summarised in Tables 2, 3 and 4.
The most striking feature of the results
is that they consistently reject the efficient forecast hypothesis.
For the
interest rate and currency options, this rejection generally occurs through
estimates of the
B
coefficients being significantly less than one.
The
interpretation of this is that changes in the estimated volatilities
overpredict subsequent movements in the true variance of prices.
typical
B
With a
coefficient of around 0.7 for the interest rate options, the average
extent of this overprediction is estimated to be about 40 per cent of the true
changes in variance, or about 20 per cent when measured in terms of standard
deviations.
The apparent biases in currency options are even larger, with the
B coefficient estimated at around 0.5 or less.
9.
TABLE 1
TESTS FOR CONDITIONAL HETEROSCEDASTICITY IN FUTURES PRICES
Dependent
Variable
(Squared
change in
futures
price)
Bonds
Bills
SPI
Notes:
Constant
Implied
2
Lagged Dependents
(]
1
2
R2
dw
4
3
.0002
(0.71)
.0001
(0.61)
0.49
( 1. 97)
0.32
( 1. 65)
0.13
(1.28)
0.08
(0.75)
-0.06
(-0.57)
(-0.04)
(-0.36)
0.38
(0.37)
0.10
(0.98)
-0.13
(-1.31)
(-0.14)
(-1.38)
0.20
2.00
0.22
1. 96
.oo
(-.08)
.00
(1.39)
1. 49
(3.61)
1.37
(4.11)
0.07
(0.75)
-0.05
(-0.60)
-0.07
(-0.83)
-0.23
(-2.67)
-0.04
(-0.37)
-0.03
(-0.35)
0.17
(1.36)
-0.11
(-1.26)
0.29
1. 92
0.37
1. 86
.0004
( • 23)
-.002
(-0.89)
0.896
(0.77)
3.34
(2.07)
-0.35
(-2.91)
-0.57
(-3.58)
-0.29
(-2.95)
-0.46
(-3.58)
0.10
(1.02)
-0.07
(-0.66)
0.13
(1.38)
-0.11
(-0.88)
0.34
2.34
0.35
2.31
This table reports estimates of the equations
For each contract, the first row of results are estimated using call
volatilities, while the second uses puts.
Figures in parentheses are t-statistics.
TABLE 2
TESTS OF THE EFFICIENT FORECAST CONDITION:
CURRENCY OPTIONS
Constant
B
Calls
0.307 x lo-3
(2.69)
.454
(8.61)
-.261
(2.69)
-.320
(4.19)
-.023
Puts
0.278
(2.26)
lo-3
.337
(6.06)
-.213
(2.11)
-.261
(2.84)
.035
Notes:
X
Lagged squared innovations
4
8
The table reports estimates of equation (9b).
are t-statistics.
Rho(4)
Figures in parentheses
10.
The above findings are fairly consistent across the interest rate and currency
options data sets, but somewhat different results are obtained for the SPI
contracts, and it is important to note that the SPI results are quite
sensitive to the inclusion or exclusion of the sharemarket crash.
Results for
the SPI options are therefore reported on both a pre-crash and a full sample
basis.
It turns out that in both samples the joint null hypothesis is
strongly rejected, but for different reasons.
Using the pre-crash sample, the
B coefficient is estimated to be substantially less than one (the point
estimates are around 0.3) which, on the face of it, suggests an even more
significant element of market overreaction than appeared in the interest rate
contracts.
The interpretation of this result is, however, clouded by the
difficulty in quantifying the effect of possible anticipations of a major
market correction on the pricing of options in the pre-crash period.
Expected
volatilities may have been influenced by such anticipations, but there is no
obvious way of testing for this because of the uniqueness of the period around
the crash.
In the full sample, the results are quite different.
The
estimated B coefficient comes out much closer to one, and in the case of put
options is insignificant from one.
The joint hypothesis is nonetheless
rejected in this case by the significant negative coefficients obtained on
elements of the distributed lag on e!.
currency options results.
This feature is also observed in the
The negative sign here indicates that o! tends
to overpredict the subsequent variance when a large absolute price change has
been recently observed in the price of the underlying.
This would be
consistent with the view that market volatility estimates are excessively
influenced on average by current and recent measured volatility in the price
of the underlying.
This result is equally true for puts and calls.
Once again, however, there are some difficulties of interpretation, because
these results are dominated by the effects of the crash and its immediate
aftermath.
Following the crash, realised share price volatilities were
actually slightly lower on average than they had been before it, yet the
expected volatilities implied in both call and put prices remained
historically very high (see Figure 3 where these volatilities are
illustrated).
Ex post, therefore, a forecasting rule which puts a negative
weight on large recent absolute price movements is seen to produce estimates
which are much closer to the true volatility.
Given the unprecedented nature
of this period, however, it seems likely that such a rule would take some time
to be learned.
Taken together, these considerations suggest that any
11.
TABLE 3
TESTS OF THE EFFICIENT FORECAST CONDITION:
Constant
0.5 x lo-4
(3.34)
o.8 x lo-5
(0.66)
Bills
0.5 x 10-6
(3.94)
0.1 x 1o-6
(1.01)
SPI
o.6 x lo-3
(5.40)
0.4 X 103
(4.38)
Notes:
0.6989
(9.20)
1
0.6224
(8.20)
1
0.8048
(12.49)
1
dw
Lagged squared innovations
B
1
Bonds
FUTURES OPTIONS (CALLS)
2
3
4
0.05
(1.33)
0.03
(0.85)
-0.05
(-1.50)
0.04
(1.39)
0.71
2.01
0.02
(0.50)
0.01
(0.21)
-0.08
(-2.23)
0.03
(0.86)
0.66
2.18
0.001
(0.06)
0.003
(0.14)
0.03
(1.19)
0.72
1.91
-0.009
(-0.40)
-0.0008
(-0.04)
-0.006
(-0.31)
0.19
(0.63)
0.66
2.15
0.0005
(0.03)
-0.17
(-23.03)
-0.13
(-21.00)
0.03
(5.01)
0.04
(7.79)
0.95
2.00
-0.19
(-29.46)
-0.14
(-22.68)
0.02
(3.92)
0.03
(6.93)
0.95
2.26
The table reports estimates of equation (6) for call options.
Figures in parentheses are t-statistics.
TABLE 4
TESTS OF THE EFFICIENT FORECAST CONDITION:
Constant
B
1
Bonds
0.1 x lo-3
(3.96)
0.2 x lo-4
(1.08)
Bills
0.6 x lo-6
(4.87)
o.4 x 1o-6
(3.76)
SPI
o.5 x lo-3
(4.76)
o.4 x lo-3
(6.30)
Notes:
0.5268
(5.68)
1
0.7486
(8.84)
1
0.9290
(12.30)
1
FUTURES OPTIONS (PUTS)
Lagged squared innovations
2
3
4
dw
0.05
(0.91)
0.002
(0.03)
-0.02
(-0.35)
0.009
(0.18)
0.61
2.17
-0.05
(-0.88)
-0.06
(-1.00)
-0.08
(-1.34)
0.007
(0.12)
0.51
2.53
-0.02
(-0.78)
-0.04
(-2.21)
-0.03
(-1.51)
-0.01
(-0.58)
0.61
1.85
-0.03
(-1.79)
-0.05
(-3.07)
-0.04
(-2.12)
-0.02
(-1.04)
0.58
2.05
-0.185
(-25.16)
-0.17
(-29.32)
0.03
(6.88)
-0.001
(-0.25)
0.97
1.91
-0.19
(-41.50)
-0.17
(-38.18)
0.03
(8.27)
-0.004
(-1.33)
0.97
1.99
The table reports estimates of equation (6) for put options.
Figures in parentheses are t-statistics.
12.
conclusions concerning apparent biases in SPI options prices, need to be
heavily qualified.
5.
Further results on pricing discrepancies
It has been pointed out that results of the kind reported in section 4 are
necessarily conditional on the assumed option pricing model;
rejection of the
joint null hypothesis might, therefore, indicate rejection of the
Black-Scholes model rather than rejection of forecast efficiency.
This
section considers two additional pieces of evidence which may have a bearing
on this question.
The first uses the fact that when the Black-Scholes
assumptions are violated, the bias in the formula will be related to the
extent to which the option being priced is in or out of the money;
the
formula tends to overvalue deep in-the-money options and to undervalue those
which are deep out-of-the-money (see for example, Merton (1976)).
Thus, if
the apparent pricing biases detected in section 4 are caused partially by
deviations from the Black-Scholes assumptions, we would expect the empirical
biases to be correlated with the option deltas, which can be used as a measure
of the extent to which an option is in or out of the money.
Estimated pricing biases can be obtained by using fitted values for expected
volatility from equations (6) and (Sb) to obtain a time series of estimated
option values, which can then be compared with observed prices.
A comparison
between the estimated volatilities and those implied in the option prices
appears in Figures 1 to 4.
Generally speaking, these imply price
discrepancies which vary in a range between zero and about 30 per cent.
(For
the SPI options the calculations, though performed for the full sample, are
based on the estimated pre-crash relationship.)
The graphs also illustrate
the general point that "fitted" volatilities tend to be less variable than
those implied in actual option prices.
The currency options case differs from
the other three in that there is also an apparent unidirectional bias in
levels, with implied volatilities always exceeding those predicted by the
forecasting equation.
However, the gap narrows through the sample period,
perhaps reflecting an increasing competitiveness in the pricing of these
options.
13.
IMPLIED AND PREDICTED VOLATILITIES
Figure 1
10 YEAR BONDS CALL OPTIONS
-Implied
o;o
Ul~
Predicted
18
16
~A
14
12
~
A
10
~'
~~
/.
8
6
4
2
0
1 986
1987
1 988
Figure 2
90 DAY BANK BILLS CALL OPTIONS
-Implied
o;o
'""' Predicted
2
1. 8
1. 6
1 .4
1 .2
r.~
0.8
'v
F
0.6
0.4
0.2
JuJ
r~
0
1 986
1987
1 988
14.
IMPLIED AND PREDICTED VOLATILITIES
Figure 3
SHARE PRICE INDEX CALL OPTIONS
-Implied
•En
Predicted
0
/o
80
70
60
50
40
30
20
10
0
1987
1986
1988
Figure 4
CURRENCY OPTIONS
-
Implied
•Fa
Predicted
45
40
35
30
25
10
5
o~--------L---------~----------~--------~-----------L---
Mar-85
Sep-85
Mar-86
Sep-86
Mar-87
Sep-87
15.
The proposition that the pricing biases are related to the options' deltas is
tested using simple linear regressions of the calculated biases against the
relevant series of deltas.
(This is not applicable to the case of currency
options, since all the currency options in the data set are at-the-money.)
Results for these regressions are reported in Table 5.
They show quite
clearly a lack of any significant correlation between these variables;
estimated pricing biases appear not to be correlated with the extent to which
options are close to the money.
This provides one piece of evidence that our
results are unlikely to be explained as being due to deviations from the
Black-Scholes assumptions.
As a second piece of evidence, which has a direct bearing on market
efficiency, it may be asked whether use of statistically efficient forecasts
implied by use of equations (6) and (Bb) would have led to significant excess
returns within the sample period.
(At a later date it is hoped to repeat this
exercise using out of sample data.)
To investigate this question the
comparison between actual and estimated option prices described above was used
to simulate the following hedged trading rule:
buy any option that is undervalued and hold to expiry;
sell any option that is overvalued, and hold the position to expiry;
in each case, delta hedge the initial position in the underlying
instrument, but with no subsequent rebalancing.
The delta is evaluated using fitted volatilities.
The realised excess return on a hedged call transaction under this rule is
then given by:
rr = D
where
{-c + max (fT - x, 0) - &(f
D
= 1
= -1
T
- f )}
t
for a buy
for a sell
Put returns are defined analogously.
Within sample returns calculated in this way are summarised in Table 6.
(9)
16.
TABLE 5
CORRELATIONS BETWEEN ESTIMATED PRICE BIASES AND OPTION DELTAS
Constant
Option Delta
R2
dw
0.09
(4.19)
-0.006
(-0.30)
0.50
1.05
0.16
(4.62)
0.04
( 1. 13)
0.15
0.95
0.12
(4.01)
0.06
(2.11)
0.12
1.38
puts
0.16
(7.18)
-0.02
(-1.04)
0.24
1.04
calls
0.12
(2.38)
0.11
( 1. 64)
0.32
1.84
puts
0.07
(2.38)
0.04
(1.31)
0.26
0.92
De~endent variable
Pr~ce
bias on:
Bonds calls
puts
Bills calls
SPI
Note:
t - statistics in parentheses.
TABLE 6
PROPORTION OF OBSERVATIONS SHOWING WITHIN-SAMPLE EXCESS RETURNS
Calls
Note:
Puts
Bonds
58
57
Bills
70
76
SPI
54
51
Currency
77
80
The table shows the percentage of observations in the sample showing
excess returns as defined by equation ( 10) •
17.
The most dramatic results are those for bill futures options, and currency
options.
In both these cases roughly three-quarters of the observations
generate positive excess returns within sample.
The currency option results
are generated mainly by an implicit overpricing during the sample period;
since forecast volatilities were always less than those implied in option
prices, the trading rule given by equation (9) indicated a sell in every
period.
For bond and SPI options the figures for excess returns are smaller
at around 60 per cent for bonds, while for the SPI, it is barely above the
50 per cent which one would expect to obtain randomly.
This perhaps confirms
some of the reservations which were expressed in the earlier discussion of the
SPI results.
For the remainder, however, the results provide some support for
the view that expectations errors, rather than biases in the pricing formula,
are responsible for the anomalies reported in section 4.
In this regard, it
is worth noting that the rule used to detect excess returns here was a
particularly unsophisticated one.
Outstanding options were not assumed to be
delta hedged to maturity, and there was no attempt to weight the size of each
trade according to the relative extent of each overvaluation or
undervaluation.
If anything, such a trading rule would be likely to
understate the presence of excess returns in the sample.
6.
Conclusions
The paper claims to have detected a tendency for both financial futures
options and currency options to overpredict subsequent changes in the variance
of the underlying prices.
In currency options, the results also imply a
consistent overstatement of the level of volatility, though this has tended to
narrow steadily through the sample period.
These anomalies were statistically
significant in all the cases tested, but we recognise that the unusual
circumstances surrounding recent share price movements make the results for
SPI options difficult to interpret.
For the remainder, however, the results
clearly violate the joint hypothesis of Black-Scholes pricing and forecast
efficiency.
The fact that apparent pricing biases are not correlated with
option deltas, and give rise to within-sample excess returns, points to
forecast inefficiencies as an important element in explaining these results.
18.
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